\(\int \frac {(a+b x) (a c-b c x)^5}{x^{10}} \, dx\) [41]

Optimal result

Integrand size = 20, antiderivative size = 82 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{10}} \, dx=-\frac {a^6 c^5}{9 x^9}+\frac {a^5 b c^5}{2 x^8}-\frac {5 a^4 b^2 c^5}{7 x^7}+\frac {a^2 b^4 c^5}{x^5}-\frac {a b^5 c^5}{x^4}+\frac {b^6 c^5}{3 x^3} \]

[Out]

-1/9*a^6*c^5/x^9+1/2*a^5*b*c^5/x^8-5/7*a^4*b^2*c^5/x^7+a^2*b^4*c^5/x^5-a*b^5*c^5/x^4+1/3*b^6*c^5/x^3

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {76} \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{10}} \, dx=-\frac {a^6 c^5}{9 x^9}+\frac {a^5 b c^5}{2 x^8}-\frac {5 a^4 b^2 c^5}{7 x^7}+\frac {a^2 b^4 c^5}{x^5}-\frac {a b^5 c^5}{x^4}+\frac {b^6 c^5}{3 x^3} \]

[In]

Int[((a + b*x)*(a*c - b*c*x)^5)/x^10,x]

[Out]

-1/9*(a^6*c^5)/x^9 + (a^5*b*c^5)/(2*x^8) - (5*a^4*b^2*c^5)/(7*x^7) + (a^2*b^4*c^5)/x^5 - (a*b^5*c^5)/x^4 + (b^
6*c^5)/(3*x^3)

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] &&  !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^6 c^5}{x^{10}}-\frac {4 a^5 b c^5}{x^9}+\frac {5 a^4 b^2 c^5}{x^8}-\frac {5 a^2 b^4 c^5}{x^6}+\frac {4 a b^5 c^5}{x^5}-\frac {b^6 c^5}{x^4}\right ) \, dx \\ & = -\frac {a^6 c^5}{9 x^9}+\frac {a^5 b c^5}{2 x^8}-\frac {5 a^4 b^2 c^5}{7 x^7}+\frac {a^2 b^4 c^5}{x^5}-\frac {a b^5 c^5}{x^4}+\frac {b^6 c^5}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.83 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{10}} \, dx=c^5 \left (-\frac {a^6}{9 x^9}+\frac {a^5 b}{2 x^8}-\frac {5 a^4 b^2}{7 x^7}+\frac {a^2 b^4}{x^5}-\frac {a b^5}{x^4}+\frac {b^6}{3 x^3}\right ) \]

[In]

Integrate[((a + b*x)*(a*c - b*c*x)^5)/x^10,x]

[Out]

c^5*(-1/9*a^6/x^9 + (a^5*b)/(2*x^8) - (5*a^4*b^2)/(7*x^7) + (a^2*b^4)/x^5 - (a*b^5)/x^4 + b^6/(3*x^3))

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.74

method	result	size
gosper	\(-\frac {c^{5} \left (-42 b^{6} x^{6}+126 a \,x^{5} b^{5}-126 a^{2} x^{4} b^{4}+90 a^{4} x^{2} b^{2}-63 a^{5} x b +14 a^{6}\right )}{126 x^{9}}\)	\(61\)
default	\(c^{5} \left (-\frac {5 a^{4} b^{2}}{7 x^{7}}+\frac {a^{5} b}{2 x^{8}}+\frac {b^{6}}{3 x^{3}}-\frac {a \,b^{5}}{x^{4}}+\frac {a^{2} b^{4}}{x^{5}}-\frac {a^{6}}{9 x^{9}}\right )\)	\(61\)
norman	\(\frac {a^{2} b^{4} c^{5} x^{4}-\frac {1}{9} a^{6} c^{5}+\frac {1}{3} b^{6} c^{5} x^{6}-a \,b^{5} c^{5} x^{5}-\frac {5}{7} a^{4} b^{2} c^{5} x^{2}+\frac {1}{2} a^{5} b \,c^{5} x}{x^{9}}\)	\(74\)
risch	\(\frac {a^{2} b^{4} c^{5} x^{4}-\frac {1}{9} a^{6} c^{5}+\frac {1}{3} b^{6} c^{5} x^{6}-a \,b^{5} c^{5} x^{5}-\frac {5}{7} a^{4} b^{2} c^{5} x^{2}+\frac {1}{2} a^{5} b \,c^{5} x}{x^{9}}\)	\(74\)
parallelrisch	\(\frac {42 b^{6} c^{5} x^{6}-126 a \,b^{5} c^{5} x^{5}+126 a^{2} b^{4} c^{5} x^{4}-90 a^{4} b^{2} c^{5} x^{2}+63 a^{5} b \,c^{5} x -14 a^{6} c^{5}}{126 x^{9}}\)	\(76\)

[In]

int((b*x+a)*(-b*c*x+a*c)^5/x^10,x,method=_RETURNVERBOSE)

[Out]

-1/126*c^5*(-42*b^6*x^6+126*a*b^5*x^5-126*a^2*b^4*x^4+90*a^4*b^2*x^2-63*a^5*b*x+14*a^6)/x^9

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{10}} \, dx=\frac {42 \, b^{6} c^{5} x^{6} - 126 \, a b^{5} c^{5} x^{5} + 126 \, a^{2} b^{4} c^{5} x^{4} - 90 \, a^{4} b^{2} c^{5} x^{2} + 63 \, a^{5} b c^{5} x - 14 \, a^{6} c^{5}}{126 \, x^{9}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^10,x, algorithm="fricas")

[Out]

1/126*(42*b^6*c^5*x^6 - 126*a*b^5*c^5*x^5 + 126*a^2*b^4*c^5*x^4 - 90*a^4*b^2*c^5*x^2 + 63*a^5*b*c^5*x - 14*a^6
*c^5)/x^9

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{10}} \, dx=- \frac {14 a^{6} c^{5} - 63 a^{5} b c^{5} x + 90 a^{4} b^{2} c^{5} x^{2} - 126 a^{2} b^{4} c^{5} x^{4} + 126 a b^{5} c^{5} x^{5} - 42 b^{6} c^{5} x^{6}}{126 x^{9}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)**5/x**10,x)

[Out]

-(14*a**6*c**5 - 63*a**5*b*c**5*x + 90*a**4*b**2*c**5*x**2 - 126*a**2*b**4*c**5*x**4 + 126*a*b**5*c**5*x**5 -
42*b**6*c**5*x**6)/(126*x**9)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{10}} \, dx=\frac {42 \, b^{6} c^{5} x^{6} - 126 \, a b^{5} c^{5} x^{5} + 126 \, a^{2} b^{4} c^{5} x^{4} - 90 \, a^{4} b^{2} c^{5} x^{2} + 63 \, a^{5} b c^{5} x - 14 \, a^{6} c^{5}}{126 \, x^{9}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^10,x, algorithm="maxima")

[Out]

1/126*(42*b^6*c^5*x^6 - 126*a*b^5*c^5*x^5 + 126*a^2*b^4*c^5*x^4 - 90*a^4*b^2*c^5*x^2 + 63*a^5*b*c^5*x - 14*a^6
*c^5)/x^9

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{10}} \, dx=\frac {42 \, b^{6} c^{5} x^{6} - 126 \, a b^{5} c^{5} x^{5} + 126 \, a^{2} b^{4} c^{5} x^{4} - 90 \, a^{4} b^{2} c^{5} x^{2} + 63 \, a^{5} b c^{5} x - 14 \, a^{6} c^{5}}{126 \, x^{9}} \]

[In]

integrate((b*x+a)*(-b*c*x+a*c)^5/x^10,x, algorithm="giac")

[Out]

1/126*(42*b^6*c^5*x^6 - 126*a*b^5*c^5*x^5 + 126*a^2*b^4*c^5*x^4 - 90*a^4*b^2*c^5*x^2 + 63*a^5*b*c^5*x - 14*a^6
*c^5)/x^9

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.90 \[ \int \frac {(a+b x) (a c-b c x)^5}{x^{10}} \, dx=-\frac {\frac {a^6\,c^5}{9}-\frac {a^5\,b\,c^5\,x}{2}+\frac {5\,a^4\,b^2\,c^5\,x^2}{7}-a^2\,b^4\,c^5\,x^4+a\,b^5\,c^5\,x^5-\frac {b^6\,c^5\,x^6}{3}}{x^9} \]

[In]

int(((a*c - b*c*x)^5*(a + b*x))/x^10,x)

[Out]

-((a^6*c^5)/9 - (b^6*c^5*x^6)/3 + a*b^5*c^5*x^5 + (5*a^4*b^2*c^5*x^2)/7 - a^2*b^4*c^5*x^4 - (a^5*b*c^5*x)/2)/x
^9